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Mirrors > Home > ILE Home > Th. List > nltmnf | GIF version |
Description: No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.) |
Ref | Expression |
---|---|
nltmnf | ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfnre 7977 | . . . . . . 7 ⊢ -∞ ∉ ℝ | |
2 | 1 | neli 2444 | . . . . . 6 ⊢ ¬ -∞ ∈ ℝ |
3 | 2 | intnan 929 | . . . . 5 ⊢ ¬ (𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) |
4 | 3 | intnanr 930 | . . . 4 ⊢ ¬ ((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <ℝ -∞) |
5 | pnfnemnf 7989 | . . . . . 6 ⊢ +∞ ≠ -∞ | |
6 | 5 | nesymi 2393 | . . . . 5 ⊢ ¬ -∞ = +∞ |
7 | 6 | intnan 929 | . . . 4 ⊢ ¬ (𝐴 = -∞ ∧ -∞ = +∞) |
8 | 4, 7 | pm3.2ni 813 | . . 3 ⊢ ¬ (((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) |
9 | 6 | intnan 929 | . . . 4 ⊢ ¬ (𝐴 ∈ ℝ ∧ -∞ = +∞) |
10 | 2 | intnan 929 | . . . 4 ⊢ ¬ (𝐴 = -∞ ∧ -∞ ∈ ℝ) |
11 | 9, 10 | pm3.2ni 813 | . . 3 ⊢ ¬ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ)) |
12 | 8, 11 | pm3.2ni 813 | . 2 ⊢ ¬ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ))) |
13 | mnfxr 7991 | . . 3 ⊢ -∞ ∈ ℝ* | |
14 | ltxr 9749 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (𝐴 < -∞ ↔ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ))))) | |
15 | 13, 14 | mpan2 425 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 < -∞ ↔ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ))))) |
16 | 12, 15 | mtbiri 675 | 1 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 ℝcr 7788 <ℝ cltrr 7793 +∞cpnf 7966 -∞cmnf 7967 ℝ*cxr 7968 < clt 7969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-xp 4628 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 |
This theorem is referenced by: mnfle 9766 xrltnsym 9767 xrlttr 9769 xrltso 9770 xltnegi 9809 xposdif 9856 qbtwnxr 10231 xrmaxiflemab 11226 xrmaxltsup 11237 xrbdtri 11255 blssioo 13678 |
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